Question: A weight is attached by a string to the end of a spring and is thrown upwards while a stopwatch is started at $t=0$ seconds. The weight starts oscillating vertically in a periodic way that can be modeled by a trigonometric function. The weight reaches a maximum height of $12 \text{ cm}$ at $t=1.5$ seconds and falls to a minimum height of $4 \text{ cm}$ before returning to its maximum height at $t=6.5$ seconds. Find the formula of the trigonometric function that models the height $H$ of the weight $t$ seconds after it was thrown upwards. Define the function using radians. $ H(t) = $ $ $
Solution: Let's start by writing a formula for the height of the weight $u$ seconds after its peak. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. Since the weight is at its peak at time $u = 0$, let's use a cosine function to model its distance, since cosine functions also reach a peak at $u = 0$. The height of the weight has period $5$ seconds. Its midline is halfway between its maximum of $12 \text{ cm}$ and its minimum of $4\text{ cm}$, or $\dfrac{12 + 4}{2} = 8$ Its amplitude is half the difference between its maximum of $12 \text{ cm}$ and its minimum of $4 \text{ cm}$, or $\dfrac{12 - 4}{2} = 4$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we can stretch it horizontally by a factor of ${\dfrac{5}{2\pi}}$, stretch it vertically by a factor of ${4}$, and move it up ${8}$ units: $ H(u) = {4}\cos\left({\dfrac{2\pi}{5}}u\right) + {8}$ Since the weight reaches its peak $1.5$ seconds after the stopwatch is started, $t$ seconds after it's released is $t - 1.5$ seconds after that peak, so $u = t - 1.5$. $ H(t) = {4}\cos\left({\dfrac{2\pi}{5}}(t - 1.5)\right) + {8}$ The function $ H(t) = {4}\cos\left({\dfrac{2\pi}{5}}(t - 1.5)\right) + {8}$ has period $5$, amplitude $4$, and midline $y = 8$, and it reaches its peak at time $t= 1.5$, so it's a good model of the height of the weight.